# Preprint: Hausdorff Dimension of the Contours of Symmetric Additive Lévy Processes

## Davar Khoshnevisan, Narn-Rueih Shieh, and Yimin Xiao

Abstract. Let $$X_1\,,\ldots,X_N$$ denote N independent, symmetric Lévy processes on $${\bf R}^d.$$The corresponding additive Lévy process is defined as the following $$N$$-parameter random field on $${\bf R}^d$$:
$$X({\bf t}) := X_1(t_1)+\cdots+X_N(t_N)\hskip1in ({\bf t}\in{\bf R}^N_+).$$ Khoshnevisan and Xiao (2002) have found a necessary and sufficient condition for the zero-set $$X^{-1}\{0\}$$ of $$X$$ to be non-trivial with positive probability. They also provide bounds for the Hausdorff dimension of $$X^{-1}\{0\}$$ which hold with positive probability in the case that $$^{-1}\{0\}$$ can be non void. Here, we prove that the Hausdorff dimension of $$X^{-1}\{0\}$$ is a constant almost surely on the event $$\{X^{-1}\{0\}\cap F\neq\emptyset\}$$. Moreover, we derive a formula for the said constant. This portion of our work extends the one-parameter formulas of Horowitz (1968) and Hawkes (1974). More generally, we prove that for every non-random Borel set $$F$$ in $$(0\,,\infty)^N$$, the Hausdorff dimension of $$X^{-1}\{0\}\cap F$$ is a constant almost surely on the event $$\{X^{-1}\{0\}\cap F\neq\emptyset\}$$. This constant is computed explicitly in many cases.

Keywords. Additive Lévy processes, level sets, Hausdorff dimension

AMS Classification (2000) Primary. 60G70 Secondary. 60F15

Support. Research supported in part by a grant from the National Science Foundation grant DMS-0404729.

Pre/E-Prints. This paper is available in

 Davar Khoshnevisan Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. davar@math.utah.edu Narn-Rueih Shieh Department of Mathematics, National Taiwan University Taipei 10617, Taiwan shiehnr@math.ntu.edu.tw Yimin Xiao Department of Statistics and Probability, A-413 Wells Hall Michigan State University East Lansing, MI 48824, USA xiao@stt.msu.edu

Last Update: September 25, 2008
© 2006 - Davar Khoshnevisan, Narn-Rueih Shieh, and Yimin Xiao